Archimedes Principal

Is the crown pure gold?


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Archimedes was born in 287 BC in a Greek city of Syracuse in what is now Sicily.  He was a mathematician and scientist.  The king of the city, named Hiero II, provided a quantity of pure gold for the fashioning of a laurel wreath made of gold.  It was meant as an offering to the gods of the city.  The goldsmith presented the king with his crown.  It looked magnificent, but to be sure the smith had not kept any of the gold for himself, the king weighed the crown and determined it weighed exactly the same as the pure gold given him for the project.  So the smith was paid.

Later the king was informed that if the smith had replaced a small amount of the pure gold with an equal weight of silver and mixed the gold and silver, the product would look like pure gold.  Not only was the king angered that the smith would cheat him, he worried that if the crown were not pure gold, but it would also offend the gods and bring misfortune to the city.

So he contacted Archimedes and tasked him with determining if the gold crown was indeed pure gold.  Archimedes knew that pure gold would be denser than gold mixed with silver.  All he had to do was to determine the density of the crown and he would have the king’s answer.  But he had to figure out how to do this without melting down the crown to determine its volume.

The story goes that he was contemplating this problem and decided to think about it while taking a bath. After the tub was full of warm water, he stepped into it and as he lowered himself into the water, the water level rose and some of the water overflowed the tub.  This gave him an idea on how to solve the problem.  Allegedly he was so excited he ran naked through the streets to his study to begin conducting his tests, yelling I have it, which in Greek would be Eureka!  Whether this last part is true, the secret to measuring the density of the crown (or any other oddly shaped object) is to submerge it in water.




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Your Assignment

Using information that would have been available to Archimedes at the time, you are going to perform some simple mass density calculations to determine the validity of his principle.  In other words, you are going to determine for yourself whether or not the crown was made of pure gold.

Archimedes’ Principle is based upon the approach that if you know the mass of an object, and you can determine the volume of water that it displaces, you can find the density of the object. Today, we calculate density as follows:

Density=Mass /Volume































Unit 3 Case Study: Archimedes and the Gold Crown


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The challenge is to determine if a gold crown is pure gold or gold mixed with another metal. There are three ways to do this.

The first is to fill a pitcher to the brim with water, lower the object into the water, and catch the water that overflows in another container. Measure the volume of the water and divide it into the mass of the object to get the density.

To see for yourself what this might involve, use the following numbers to calculate the density of the crown. (While Archimedes would not use the units of grams and cubic centimeters, we’ll use those units for our calculations.)





Density Calculation Result Units
Is this pure gold?  


Early Greeks could not measure volume to the same accuracy as can be done today, so this method may not have worked for Archimedes. A more practical approach is as follows:


First, Archimedes would confirm that the crown had the mass of the gold provided originally: 3014 g.

Next, he tied the crown to a string tied to one end of the scale. Then he submerged the crown under water and found the mass required to exactly balance the scale. Let’s say the scale balanced at 2845.4 g.

The difference between the mass of the crown in air and its mass while submerged is the mass of the water displaced by the crown.

The mass of the water provides a way to calculate the volume of water, provided you know that the density of water is . Hint: rearrange the definition of Density to calculate volume.

Calculation Result Unit
Mass of water displaced =Mass of Gold – Mass of Gold under water    
Volume of water with this mass = =    
Is this pure gold?  


 A third way of doing this would be to balance the crown with an equal mass of pure gold, melted into an object and tied to the balance beam. Then lower both crown and pure gold into a water. Hint: which should displace more water, pure gold or a gold mixed with another metal?


If such a test were performed and the crown were made with gold mixed with another metal, which way would the balance tip? Explain why the balance should tip this way below.


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